Method of characterizing fiber bragg gratings using iterative processing

ABSTRACT

A method determines a complex reflection impulse response of a fiber Bragg grating. The method includes providing a measured amplitude of a complex reflection spectrum of the fiber Bragg grating. The method further includes providing an estimated phase term of the complex reflection spectrum. The method further includes multiplying the measured amplitude and the estimated phase term to generate an estimated complex reflection spectrum. The method further includes calculating an inverse Fourier transform of the estimated complex reflection spectrum, wherein the inverse Fourier transform is a function of time. The method further includes calculating an estimated complex reflection impulse response by applying at least one constraint to the inverse Fourier transform of the estimated complex reflection spectrum.

CLAIM OF PRIORITY

This application claims the benefit of U.S. Provisional Application Nos.60/571,660, filed May 15, 2004, 60/599,427, filed Aug. 6, 2004, and60/662,684, filed Mar. 17, 2005, each of which is incorporated in itsentirety by reference herein.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates generally to apparatus and methods ofcharacterizing the optical response of fiber Bragg gratings.

2. Description of the Related Art

Fiber Bragg gratings (FBGs) have many applications in opticalcommunications and optical fiber sensing. The effective refractive indexprofile Δn(z) of the fiber core mode (e.g., the LP₀₁ mode, or ahigher-order mode) as a function of the position z along the FBGgenerally varies roughly periodically with z, with an envelope that mayvary along z. The effective refractive index Δn(z) determines most ofthe optical properties of the FBG, including but not limited to, thedispersion properties, the complex reflection impulse response h_(R)(t),the complex transmission impulse response h_(T)(t), the amplitudes|r(ω)|, |t(ω)| and the phases φ_(r)(ω), φ_(t)(ω) of the complexreflection spectrum r(ω) and the complex transmission spectrum t(ω),respectively, and the group delay in reflection dφ_(r)(ω)/dω andtransmission dφ_(t)(ω)/dω, where ω is the angular frequency. Thesefunctions can also be expressed as functions of the optical wavenumberk, which has a simple relationship to the optical angular frequency ω,so it is a simple matter to switch between ω and k.

A first general method to determine Δn(z) of an FBG is to measure thecomplex reflection impulse response h_(R)(t), which is the temporaldependence of the amplitude and phase of the signal reflected by the FBGwhen an extremely short optical signal is launched into the FBG. Thecomplex reflection impulse response h_(R)(t) can be measured directly bylaunching an ultra-short pulse (e.g., approximately 1 picosecond toapproximately 30 picoseconds, depending on the grating length andperiod) into the FBG and measuring the temporal evolution of thereflected signal. This general method has the drawback of requiring thatthe width of the input laser pulse be much narrower than the impulseresponse of the FBG. In addition, interferometric techniques are used tomeasure both the phase and the amplitude of the complex reflectionimpulse response, and these techniques are complicated and inherentlysensitive to noise or other fluctuations.

A second general technique to measure the complex reflection impulseresponse h_(R)(t) is to use an interferometer to measure the wavelengthdependence of both the amplitude and the phase of the optical signalreflected by the FBG (i.e., the complex reflection spectrum r(ω) orr(k)). The complex reflection spectrum r(ω) is the Fourier transform(FT) of the complex reflection impulse response h_(R)(t), as describedby A. Rosenthal and M. Horowitz, “Inverse scattering algorithm forreconstructing strongly reflecting fiber Bragg gratings,” IEEE Journalof Quantum Electronics, Vol. 39, pp. 1018-1026, August 2003. The complexreflection impulse response h_(R)(t) is then recovered from the complexreflection spectrum r(ω) by taking the inverse Fourier transform (IFT)of r(ω). As discussed below, the main difficulty of this generaltechnique is that the measurement of the complex reflection spectrum isin general tedious, sensitive to noise, applicable to only special typesof FBGs, and/or time-consuming.

The complex reflection spectrum r(ω)=|r(ω)|·exp(jφ_(r)(ω)) of an FBG ismeasurable using various interferometric measurement systems which aregenerally more complex and have stronger noise sensitivities than domeasurement techniques which merely provide the amplitude of thereflection or transmission spectra. For example, in Michelsoninterferometry (e.g., as described by D.-W. Huang and C.-C. Yang,“Reconstruction of Fiber Grating Refractive-Index Profiles From ComplexBragg Reflection Spectra,” Applied Optics, 1999, Vol. 38, pp.4494-4499), a tunable laser and an optical spectrum analyzer (OSA) areused to recover the phase of the complex reflection spectrum from threeindependent measurements.

In end-reflection interferometry (e.g., as described by J. Skaar,“Measuring the Group Delay of Fiber Bragg Gratings by Use ofEnd-Reflection Interference,” Optics Letters, 1999, Vol. 24, pp.1020-1022), the FBG is characterized using a tunable laser together withan OSA by measuring the spectral reflectivity that is caused by theinterference between the FBG itself and the bare fiber end. Thistechnique, however, is generally a destructive technique, since the barefiber end must typically be only a few centimeters away from the FBG.

In low-coherence time reflectometry (e.g., as described by P. Lambeletet al., “Bragg Grating Characterization by Optical Low-CoherenceReflectometry,” IEEE Photonics Technology Letters, 1993, Vol. 5, pp.565-567; U. Wiedmann et al, “A Generalized Approach to OpticalLow-Coherence Reflectometry Inducing Spectral Filtering Effects,” J. ofLightwave Technol., 1998, Vol. 16, pp. 1343-1347; E. I. Petermann etal., “Characterization of Fiber Bragg Gratings by Use of OpticalCoherence-Domain Reflectometry,” J. of Lightwave Technol., 1999, vol.17, pp. 2371-2378; and S. D. Dyer et al., “Fast and AccurateLow-Coherence Interferometric Measurements of Fiber Bragg GratingDispersion and Reflectance,” Optics Express, 1999, Vol. 5, pp. 262-266),a Michelson interferometer is illuminated with a broadband light source,and light reflected from the FBG, placed on one arm of theinterferometer, and light reflected from a moveable mirror, placed onthe reference arm of the interferometer, are coupled together anddirected to a detector. This technique utilizes a slow mechanical scanto retrieve the impulse response of the FBG as a function of time, whichmakes this type of measurement time-consuming.

In low-coherence spectral interferometry (e.g., as described by S. Kerenand M. Horowitz, “Interrogation of Fiber Gratings by Use ofLow-Coherence Spectral Interferometry of Noiselike Pulses,” OpticsLetters, 2001, Vol. 26, pp. 328-330; and S. Keren et al., “Measuring theStructure of Highly Reflecting Fiber Bragg Gratings,” IEEE Photon. Tech.Letters, 2003, Vol. 15, pp. 575-577), the slow scanning process isavoided by reflecting broadband laser pulses from the FBG and temporallycombining these reflected pulses with their delayed replicas. This pulsesequence is then sent to an OSA, which records the power spectrum. Thepulsed laser source of this technique has an autocorrelation functionwhich is temporally much narrower (e.g., approaching a delta function)than the impulse response of the FBG. In other words, the recovery ofthe impulse response of a given FBG is limited in resolution to theautocorrelation trace of the pulsed laser source. Furthermore, the delaybetween the reflected pulse from the FBG and the input laser pulse hasto be carefully adjusted to avoid overlap in the inverse Fouriertransform domain, which makes the recovery impossible due to aliasing.

Typically, measurement systems which measure the amplitude of thereflection spectrum or of the transmission spectrum do not provide themissing phase information (i.e., φ_(r)(ω) and/or φ_(t)(ω)). Theamplitude measurement, which is relatively simpler than the phasemeasurement, involves a tunable laser and an optical spectrum analyzer(OSA). Previously, various methods have been proposed to reconstruct themissing phase spectrum or group delay spectrum from only the amplitudemeasurement of |r(ω)| or |t(ω)|. The phase reconstruction techniquepresented by Muriel et al., “Phase Reconstruction From Reflectivity inUniform Fiber Bragg Gratings,” Optics Letters, 1997, Vol. 22, pp. 93-95,only works for uniform gratings and has been independently shown to beunsuited for gratings with imperfections (J. Skaar and H. E. Engan,“Phase Reconstruction From Reflectivity in Fiber Bragg Gratings,” OpticsLetters, 1999, Vol. 24, pp. 136-138). A similar technique has beensuggested to improve the noise performance of the initial technique ofMuriel et al., however this technique is still limited to only uniformgratings and the processing algorithm involves adjusting of filteringparameters, which depend on the FBG being characterized (K. B. Rochfordand S. D. Dyer, “Reconstruction of Minimum-Phase Group Delay From FibreBragg Grating Transmittance/Reflectance Measurements,” ElectronicsLetters, 1999, Vol. 35, pp. 838-839).

One method of recovering the phase information from the amplitude dataof FBGs was previously described by L. Poladian, “Group-DelayReconstruction for Fiber Bragg Gratings in Reflection and Transmission,”Optics Letters, 1997, Vol. 22, pp. 1571-1573. The technique of Poladianutilized the fact that the transmission spectra of all FBGs belong tothe family of minimum-phase functions (MPF) which have their phase andamplitude related by the complex Hilbert transform. In the technique ofPoladian, using the Hilbert transformation, the phase or group delay ofFBGs is recovered from only the measurement of the amplitude of thetransmission spectrum |t(ω)|. This technique works very well but thenumerical evaluation of the principle-value Cauchy integral in theHilbert transform is not trivial and is rather noise-sensitive, asdescribed by Muriel et al.

SUMMARY OF THE INVENTION

In certain embodiments, a method determines a complex reflection impulseresponse of a fiber Bragg grating. The method comprises providing ameasured amplitude of a complex reflection spectrum of the fiber Bragggrating. The method further comprises providing an estimated phase termof the complex reflection spectrum. The method further comprisesmultiplying the measured amplitude and the estimated phase term togenerate an estimated complex reflection spectrum. The method furthercomprises calculating an inverse Fourier transform of the estimatedcomplex reflection spectrum, wherein the inverse Fourier transform is afunction of time. The method further comprises calculating an estimatedcomplex reflection impulse response by applying at least one constraintto the inverse Fourier transform of the estimated complex reflectionspectrum.

In certain embodiments, a computer system comprises means for estimatingan estimated phase term of a complex reflection spectrum of a fiberBragg grating. The computer system further comprises means formultiplying a measured amplitude of the complex reflection spectrum ofthe fiber Bragg grating and the estimated phase term to generate anestimated complex reflection spectrum. The computer system furthercomprises means for calculating an inverse Fourier transform of theestimated complex reflection spectrum, wherein the inverse Fouriertransform is a function of time. The computer system further comprisesmeans for calculating an estimated complex reflection impulse responseby applying at least one constraint to the inverse Fourier transform ofthe estimated complex reflection spectrum.

In certain embodiments, a method determines a complex transmissionimpulse response of a fiber Bragg grating. The method comprisesproviding a measured amplitude of a complex transmission spectrum of thefiber Bragg grating. The method further comprises providing an estimatedphase term of the complex transmission spectrum. The method furthercomprises multiplying the measured amplitude and the estimated phasetenn to generate an estimated complex transmission spectrum. The methodfurther comprises calculating an inverse Fourier transform of theestimated complex transmission spectrum, wherein the inverse Fouriertransform is a function of time. The method further comprisescalculating an estimated complex transmission impulse response byapplying at least one constraint to the inverse Fourier transform of theestimated complex transmission spectrum.

In certain embodiments, a computer system comprises means for estimatingan estimated phase term of a complex transmission spectrum of a fiberBragg grating. The computer system further comprises means formultiplying a measured amplitude of the complex transmission spectrum ofthe fiber Bragg grating and the estimated phase term to generate anestimated complex transmission spectrum. The computer system furthercomprises means for calculating an inverse Fourier transform of theestimated complex transmission spectrum, wherein the inverse Fouriertransform is a function of time. The computer system further comprisesmeans for calculating an estimated complex transmission impulse responseby applying at least one constraint to the inverse Fourier transform ofthe estimated complex transmission spectrum.

In certain embodiments, a method characterizes a fiber Bragg grating.The method comprises providing a measured amplitude of a Fouriertransform of a complex electric field envelope of an impulse response ofthe fiber Bragg grating. The method further comprises providing anestimated phase term of the Fourier transform of the complex electricfield envelope. The method further comprises multiplying the measuredamplitude and the estimated phase term to generate an estimated Fouriertransform of the complex electric field envelope. The method furthercomprises calculating an inverse Fourier transform of the estimatedFourier transform of the complex electric field envelope, wherein theinverse Fourier transform is a function of time. The method furthercomprises calculating an estimated electric field envelope of theimpulse response by applying at least one constraint to the inverseFourier transform of the estimated Fourier transform of the complexelectric field envelope.

In certain embodiments, a computer system comprises means for estimatingan estimated phase term of a Fourier transform of the complex electricfield envelope of an impulse response of a fiber Bragg grating. Thecomputer system further comprises means for multiplying a measuredamplitude of the Fourier transform of the complex electric fieldenvelope and the estimated phase term to generate an estimated Fouriertransform of the complex electric field envelope. The computer systemfurther comprises means for calculating an inverse Fourier transform ofthe estimated Fourier transform, wherein the inverse Fourier transformis a function of time. The computer system further comprises means forcalculating an estimated electric field envelope of the impulse responseby applying at least one constraint to the inverse Fourier transform ofthe estimated Fourier transform of the complex electric field envelope.

In certain embodiments, an apparatus characterizes at least one fiberBragg grating. The apparatus comprises a laser pulse source whichgenerates at least one input laser pulse. The apparatus furthercomprises an optical spectrum analyzer. The apparatus further comprisesa first optical path optically coupled to the laser pulse source,. Thefirst optical path comprises a pulse stretcher and an attenuator. Afirst portion of the input laser pulse propagates from the laser pulsesource and is stretched by the pulse stretcher and is attenuated by theattenuator. The apparatus further comprises a second optical pathoptically coupled to the first optical path and comprising a mirror. Afirst portion of the stretched and attenuated laser pulse from the firstoptical path is reflected from the mirror. The apparatus furthercomprises a third optical path optically coupled to the first opticalpath and comprising a first fiber Bragg grating. A second portion of thestretched and attenuated laser pulse from the first optical path isreflected from the first fiber Bragg grating. The apparatus furthercomprises a fourth optical path optically coupled to the second opticalpath, the third optical path, and the optical spectrum analyzer. Thereflected pulse from the mirror and the reflected pulse from the firstfiber Bragg grating propagate to the optical spectrum analyzer. Theapparatus further comprises a fifth optical path optically coupled tothe laser pulse source and the optical spectrum analyzer. The fifthoptical path comprises a delay line. A second portion of the input laserpulse propagates from the laser pulse source along the fifth opticalpath to the optical spectrum analyzer.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flowchart of an exemplary embodiment of a method ofdetermining the complex reflection impulse response h_(R)(t) of a FBG.

FIG. 2 is a flowchart of another exemplary embodiment of a method ofdetermining the complex reflection impulse response h_(R)(t) of a FBG.

FIG. 3 is a flowchart of another exemplary embodiment of a method ofdetermining the complex reflection impulse response h_(R)(t) of a FBG.

FIG. 4A schematically illustrates an exemplary embodiment of a method ofdetermining the complex transmission impulse response h_(T)(t) of a FBG.

FIG. 4B is a flowchart of the exemplary embodiment of the method of FIG.4A.

FIG. 5A is a plot of the reflection spectrum amplitude of aGaussian-apodized symmetric FBG as a function of wavelength.

FIG. 5B is a plot of the theoretical transmission group delay spectrum(sol;id curve) of the FBG of FIG. 5A as a function of wavelength and therecovered transmission group delay spectrum (dashed curve) without anynoise present.

FIG. 5C is a plot of the theoretical transmission group delay spectrum(solid curve) of the FBG of FIG. 5A as a function of wavelength and therecovered transmission group delay spectrum (dotted curve) with 5%uniform noise with a mean of 1.

FIG. 6A is a plot of the reflection spectrum amplitude of an asymmetricchirped FBG as a function of wavelength.

FIG. 6B is a plot of the theoretical transmission group delay spectrum(solid curve) of the FBG of FIG. 6A as a function of wavelength and therecovered transmission group delay spectrum (dashed curve) without anynoise present.

FIG. 6C is a plot of the theoretical transmission group delay spectrum(solid curve) of the FBG of FIG. 6A as a function of wavelength and therecovered transmission group delay spectrum (dotted curve) with 5%uniform noise with a mean of 1.

FIGS. 7A and 7B schematically illustrate two exemplary measurementconfigurations compatible with embodiments described herein.

FIG. 7C schematically illustrates an exemplary configuration of multipleFBGs compatible with certain embodiments described herein.

FIG. 8 schematically illustrates another exemplary measurementconfiguration compatible with certain embodiments described herein.

FIG. 9 is a block diagram of a method of characterizing an FBG inaccordance with certain embodiments described herein.

FIGS. 10A and 10B are plots of the theoretical electric field reflectioncoefficient amplitude and phase, respectively, of an exemplaryasymmetric chirped FBG.

FIG. 11 is a plot of the amplitude of the theoretical reflection impulseresponse of the FBG of FIGS. 10A and 10B.

FIGS. 12A and 12B illustrates the amplitude and phase, respectively, ofan exemplary temporal profile of the input laser pulse.

FIG. 13 is a plot of the normalized amplitude of the input laser pulsespectrum together with the reflection spectrum of the exemplary FBG ofFIGS. 10A and 10B.

FIG. 14 is a plot of the electric field amplitude (solid line) and phase(dashed line) for a pulse sequence inputted to an optical spectrumanalyzer in accordance with certain embodiments described herein.

FIG. 15A is a plot of the calculated power spectrum corresponding to thepulse sequence of FIG. 14.

FIG. 15B is a plot of an enlarged view of the same calculated powerspectrum of FIG. 15A.

FIGS. 16A and 16B show the amplitude and phase, respectively, therecovered complex reflection impulse response (dashed lines) and theoriginal complex reflection impulse response (solid lines) of the targetFBG.

FIGS. 17A and 17B are plots of the reflection coefficient amplitude andthe group delay spectra of the FBG calculated from the recovered complexreflection impulse response of FIGS. 16A and 16B.

FIGS. 18A and 18B are plots of the amplitude and phase, respectively, ofa different input laser pulse.

FIGS. 19A and 19B are plots of the amplitude and phase, respectively,for the recovered reflection coefficient amplitude and the group delayspectra of the FBG using the input laser pulse of FIGS. 18A and 18B.

FIG. 20 is a plot of the results from a series of simulations, each witha different splitting ratio between the peak amplitude of the inputlaser pulse and the peak amplitude of the reflected impulse response.

FIG. 21 is a plot of a simulated noisy power spectrum.

FIGS. 22A and 22B are plots of the amplitude and group delay spectra,respectively, of the reflection coefficient of the FBG calculated usingthe simulated noisy power spectrum of FIG. 21.

FIG. 23 is a plot of the normalized input laser pulse spectrum and thereflection spectra of two different Gaussian apodized FBGs which arecharacterized together in an exemplary embodiment.

FIGS. 24A and 24B are plots of the amplitude and phase of the temporalelectric field profile of the input laser pulse, respectively.

FIG. 25 is a plot of the amplitude and the phase of a pulse sequenceformed by time-delaying the two reflected pulses from the two FBGs ofFIG. 23, with respect to the leading dummy input laser pulse with asplitting ratio of 40.

FIG. 26 is a plot of the power spectrum for the pulse sequence of theleading dummy input laser pulse with the two reflected pulses of FIG.25.

FIG. 27A is a plot of the original (solid line) and recovered (dashedline) amplitude of the reflection coefficient of the first FBG of FIG.23.

FIG. 27B is a plot of the original (solid line) and recovered (dashedline) group delay spectra of the reflection coefficient of the first FBGof FIG. 23.

FIG. 28A is a plot of the original (solid line) and recovered (dashedline) amplitude of the reflection coefficient of the second FBG of FIG.23.

FIG. 28B is a plot of the original (solid line) and recovered (dashedline) group delay spectra of the reflection coefficient of the secondFBG of FIG. 23.

FIG. 29 is a plot of an FBG reflection spectrum (solid line) and thenormalized input laser pulse amplitude (dotted line) in an exemplaryembodiment.

FIGS. 30A and 30B are plots of the amplitude and phase, respectively, ofthe temporal profile of the input laser pulse.

FIG. 31 is a plot of amplitude of the impulse response (solid line) ofthe FBG and the amplitude of the reflected pulse (dashed line).

FIGS. 32A and 32B are plots of the amplitude and phase, respectively, ofthe normalized electric field envelope of the time-stretched laser pulsein an exemplary embodiment.

FIG. 33 is a plot of the pulse sequence of a dominant peak pulse at theleading edge followed by two weaker reflected pulses with a splittingratio of 40 in an exemplary embodiment.

FIG. 34 is a plot of the output of an optical spectrum analyzer for thepulse sequence of FIG. 33 in an exemplary embodiment.

FIG. 35A is a plot of the original (solid curve) and the recovered(dashed curve) of the amplitude of the reflection coefficient of the FBGof FIG. 29.

FIG. 35B is a plot of the original (solid curve) and the recovered(dashed curve) of the group delay spectra of the reflection coefficientof the FBG of FIG. 29.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

Certain embodiments described herein provide a simpler and lessnoise-sensitive technique based on an iterative process to recover thereflection phase, the reflection group delay, the transmission phase, orthe transmission group delay of an FBG (e.g., chirped, asymmetric,symmetric, or uniform) from only the measurement of the reflectionspectrum amplitude |r(ω)| or the transmission spectrum amplitude |t(ω)|.The measurements involved in certain embodiments are also substantiallyfaster than existing techniques. Furthermore, in certain embodiments inwhich the FBG is known to be symmetric, both the reflection group delayand the transmission group delay of the FBG are uniquely determined. Asdescribed herein, numerical simulations of certain embodimentsillustrate a noise sensitivity which is quite low.

Certain embodiments described herein are useful in computer-implementedanalyses of the optical characteristics of FBGs. The general-purposecomputers used for such analyses can take a wide variety of forms,including network servers, workstations, personal computers, mainframecomputers and the like. The code which configures the computer toperform such analyses is typically provided to the user on acomputer-readable medium, such as a CD-ROM. The code may also bedownloaded by a user from a network server which is part of a local-areanetwork (LAN) or a wide-area network (WAN), such as the Internet.

The general-purpose computer running the software will typically includeone or more input devices, such as a mouse, trackball, touchpad, and/orkeyboard, a display, and computer-readable memory media, such asrandom-access memory (RAM) integrated circuits and a hard-disk drive. Itwill be appreciated that one or more portions, or all of the code may beremote from the user and, for example, resident on a network resource,such as a LAN server, Internet server, network storage device, etc. Intypical embodiments, the software receives as an input a variety ofinformation concerning the material (e.g., structural information,dimensions, previously-measured amplitudes of reflection or transmissionspectra).

Iterative Processing Utilizing the Reflection Spectrum Amplitude

Measurements of the reflection spectrum amplitude |r(ω)| can be madewithout using an interferometric instrument. In certain embodiments,broadband light (e.g., white light) or laser light is launched into theFBG and an optical spectrum analyzer (OSA) or a monochromator is used torecord the reflected power R(ω)=|r(ω)|² or the transmitted powerT(ω)=1−R(ω) as a function of the optical angular frequency ω. In certainother embodiments, other techniques known in the art are used to measureeither R(ω) or T(ω). By taking the square root of R(ω), the reflectionspectrum amplitude |r(ω)| is directly obtained. However, the reflectionspectrum phase φ_(r)(ω) is still unknown. Certain embodiments describedherein retrieve this missing phase information from only either themeasured reflection power R(ω) or the measured transmission power T(ω),without carrying out additional measurements.

Certain embodiments described herein utilize an algorithm described byJ. R. Fienup in “Reconstruction of an Object from the Modulus of itsFourier Transform,” Optics Letters, 1978, Vol. 3, pp. 27-29. Thisalgorithm (referred to as “the Fienup algorithm” herein) is anerror-reduction algorithm that involves using a known (e.g., measured)Fourier transform amplitude spectrum of an unknown function g(t),together with known properties of this function (e.g., that it is a realfunction or a causal function), to correct an initial guess of g(t). Incertain embodiments, this correction is done iteratively.

FIG. 1 is a flowchart of an exemplary embodiment of a method 100 ofdetermining the complex reflection impulse response h_(R)(t) of an FBG.The method 100 comprises providing a measured reflection spectrumamplitude |r_(M)(k)| of the complex reflection spectrum r(k) of an FBGin an operational block 110. The method 100 further comprises providingan estimated phase term exp[jφ₀(k)] of the complex reflection spectrumr(k) in an operational block 120. The method 100 further comprisesmultiplying the measured reflection spectrum amplitude |r_(M)(k)| andthe estimated phase term exp[jφ₀(k)] to generate an estimated complexreflection spectrum r′(k) in an operational block 130. The method 100further comprises calculating an intermediate function h′(t), which isthe inverse Fourier transform (IFT) of the estimated complex reflectionspectrum r′(k), in an operational block 140. The method 100 furthercomprises calculating an estimated complex reflection impulse responseh_(R) ^(e)(t) by applying at least one constraint to the intermediatefunction h′(t) in an operational block 150.

In certain embodiments, providing the measured reflection spectrumamplitude |r_(M)(k)| in the operational block 110 comprises measuringthe reflection power spectrum |r_(M)(k)|² and taking the square root toyield the measured reflection spectrum amplitude |r_(M)(k)|. In otherembodiments, a previously-measured reflection spectrum amplitude|r_(M)(k)| is provided.

In certain embodiments, the measurement of the measured reflectionspectrum amplitude |r_(M)(k)| does not provide the phase term exp[jφ(k)]of the complex reflection spectrum r(k). In certain embodiments in whichthe method 100 is used iteratively, the choice of the initial estimatedphase term exp[jφ₀(k)] provided in the operational block 120 does notstrongly impact the convergence of the method. Therefore, in certainsuch embodiments, the initial estimated phase term is selected to beequal to a real or complex constant (e.g., exp[jφ₀(k)]=1). In certainother embodiments that utilize an IFT technique to provide a measuredphase term exp[jφ_(M)(k)], the estimated phase term exp[jφ₀(k)] isselected to be the measured phase term exp[jφ_(M)(k)]. In certain otherembodiments that utilize an IFT technique to provide a measured phaseterm exp[jφ_(M)(k)], the estimated phase term exp[jφ₀(k)] is selected tobe the phase term measured from another FBG which is similar to the FBGfrom which the reflection spectrum amplitude |r_(M)(k)| is measured(e.g., the FBG being characterized). By providing an estimated phaseterm which is closer to the actual phase term, certain embodimentsadvantageously reduce the convergence time of the calculations.

In the operational block 130, the measured reflection spectrum amplitude|r_(M)(k)| and the estimated phase term exp[jφ₀(k)] are multipliedtogether to generate an estimated complex reflection spectrum r′(k). Incertain embodiments, the estimated complex reflection spectrumr′(k)=|r_(M)(k)|·exp[jφ₀(k)] is a complex quantity that is calculatednumerically.

In the operational block 140, the IFT of the estimated complexreflection spectrum r′(k)=|r_(M)(k)|·exp[jφ₀(k)] is calculatednumerically. In certain embodiments, in the operational block 150, theestimated complex reflection impulse response h_(R) ^(e)(t) iscalculated by applying at least one constraint to the IFT of theestimated complex reflection spectrum r′(k). Various characteristics ofthe complex reflection impulse response h_(R) ^(e)(t) may be used assources for the applied constraint. For example, in certain embodimentsin which the complex reflection impulse response h_(R) ^(e)(t) iscausal, the t≧0 portion of the IFT of the estimated complex reflectionspectrum r′(k) is retained while the t<0 portion is set equal to zero,thereby generating an estimated complex reflection impulse responseh_(R) ^(e)(t) which is causal. In certain other embodiments, ananti-causal (or maximum-phase) constraint is applied to the IFT of theestimated complex reflection spectrum r′(k) (e.g., the t<0 portion isset equal to zero and is time-reversed). In certain embodiments in whichthe complex reflection impulse response h_(R) ^(e)(t) has a knowntemporal duration (e.g., 300 picoseconds), the portion of the IFT of theestimated complex reflection spectrum r′(k) for times greater than thetemporal duration is set equal to zero. In certain embodiments in whichthe FBG has a known bandwidth (e.g., 50 nanometers at wavelengths around1550 nanometers), then the IFT of the estimated complex reflectionspectrum r′(k) is adjusted to provide this known bandwidth. In othercertain embodiments in which the FBG is generally uniform, that is, theFBG has an effective refractive index profile which is a periodicfunction with a constant (e.g., rectangular) envelope, the complexreflection spectrum r(k) is a symmetric function, which implies that thereflection impulse response h_(R)(t) is a real function. In certain suchembodiments, the applied constraint comprises using only the realportion of the reflection impulse response h_(R) ^(e)(t). As describedmore fully below, in certain embodiments in which the calculation isiterated, the application of such constraints advantageously reduces thenumber of iterations which achieve convergence.

FIG. 2 is a flowchart of another exemplary embodiment of a method 200 ofdetermining the complex reflection impulse response h_(R)(t) of a FBG inaccordance with embodiments described herein. The method 200 comprisesthe operational blocks 110, 120, 130, 140, and 150, as described herein.The method 200 further comprises calculating a Fourier transformr_(n)(k) of the estimated complex reflection impulse response h_(R)^(e)(t) in an operational block 210. The method 200 further comprisescalculating a phase term exp[jφ_(n)(k)] of the Fourier transformr_(n)(k) of the estimated complex reflection impulse response h_(R)^(e)(t) in an operational block 220.

In certain embodiments, the Fourier transform r_(n)(k) is calculatednumerically in the operational block 210. In certain embodiments, thecalculated phase term exp[jφ_(n)(k)] of this Fourier transform r_(n)(k)is calculated numerically in the operational block 220. In certain otherembodiments in which the complex reflection spectrum is a minimum-phasefunction, as described more fully below, the calculated phase termexp[jφ_(n)(k)] of this Fourier transform r_(n)(k) is calculatedanalytically in the operational block by using a Hilbert transformationof the Fourier transform of the estimated complex reflection spectrumamplitude.

FIG. 3 is a flowchart of another exemplary embodiment of a method 300 ofdetermining the complex reflection impulse response h_(R)(t) of a FBG inaccordance with embodiments described herein. The method 300 comprisesthe operational blocks 110, 120, 130, 140, 150, 210, and 220, asdescribed herein. The method 300 further comprises using the calculatedphase term exp[jφ_(n)(k)] as the estimated phase term in the operationalblock 130 and repeating the operational blocks 130, 140, 150, 210, and220. This repeat operation is denoted in FIG. 3 by the arrow 310. Incertain such embodiments, the calculated phase term exp[jφ_(n)(k)]provides a new estimate for the missing phase term of the complexreflection spectrum. The resulting estimated Fourier transform of theoperational block 130 is the product of a measured reflection spectrumamplitude |r_(M)(k)| and a calculated estimated phase termexp[jφ_(n)(k)]. By repeating the operational blocks 130, 140, 150, 210,and 220, a second estimated complex reflection impulse response and asecond estimated phase term are generated.

In certain embodiments, the operational blocks 130, 140, 150, 210, 220as shown in FIG. 3 are iteratively repeated a number of times. Incertain such embodiments, the iterations are performed until theresulting estimated complex reflection impulse response converges.Convergence is reached in certain embodiments when the averagedifference between the estimated complex reflection impulse responsespectra obtained after two consecutive iterations is less than apredetermined value (e.g., 0.1% of the estimated complex reflectionimpulse response of the iteration). For example, convergence is reachedwhen the difference between two consecutive estimates of the function∫|r_(n)(k)−r_(n-1)(k)|² dt/∫|r_(n)(k)|² dt is less than thepredetermined value. In other embodiments, the iterations are performeda predetermined number of times (e.g., 100 times) rather thandetermining the differences between successive iterations.

In certain embodiments, the predetermined number is selected to besufficiently large such that convergence is essentially always achievedafter this number of iterations. In certain such embodiments, thepredetermined number is determined by evaluating the rate of convergencefor a number of FBGs. After a number of iterations, certain embodimentsyield an estimated phase term which is a more accurate estimate of theactual phase term than is the initial estimated phase tenn.

In certain embodiments, the number of iterations can be reduced by usingan initial estimated phase term of the complex reflection spectrum whichmore closely approximates the actual phase term of the complexreflection spectrum. For example, in certain embodiments in which thecomplex reflection spectrum is a minimum-phase function, a Hilberttransformation of the Fourier transform of the complex reflectionspectrum amplitude is used as the initial estimated phase termexp[jφ₀(k)]. In certain embodiments, only a single iteration is used tocalculate the complex reflection impulse response of the FBG. In certainembodiments, the method further comprises calculating the effectiverefractive index profile Δn(z) as a function of the position z along theFBG using the complex reflection spectrum r(k) resulting from thecalculation. Typically for non-symmetric FBGs, determining the effectiverefractive index profile Δn(z) utilizes both the complex reflectionspectrum r(k) and the complex transmission spectrum t(k). However, incertain embodiments in which the FBG is symmetric, either the complexreflection spectrum r(k) or the complex transmission spectrum t(k) isobtained using the other. Exemplary methods of calculating the effectiverefractive index profile in accordance with embodiments described hereinare described by A. Othonos and K. Kalli, “Fiber Bragg gratings:fundamentals and applications in telecommunications and sensing,” 1999,Artech House, Boston; and R. Kashyap, “Fiber Bragg gratings,” 1999,Academic Press, San Diego.

Iterative Processing Utilizing the Transmission Spectrum Amplitude

FIG. 4A schematically illustrates an exemplary embodiment of a method400 of determining the complex transmission impulse response h_(T)(t) ofa FBG in accordance with embodiments described herein. FIG. 4B is a flowdiagram of the method 400. The method 400 comprises providing a measuredtransmission spectrum amplitude |t_(M)(ω)| of the complex transmissionspectrum t(ω) of an FBG in an operational block 410. In certainembodiments in which the FBG is substantially lossless, the measuredtransmission spectrum amplitude |t_(M)(ω)| is derived from a measurementof the measured reflection spectrum amplitude |r_(M)(ω)| using therelation |t_(M)(ω)|²+|r_(M)(ω)|²=1.

The method 400 further comprises providing an estimated phase termexp[jφ₀(ω)] of the complex transmission spectrum t(ω) in an operationalblock 420. In certain embodiments, the phase term exp[jφ₀(ω)] of thecomplex transmission spectrum t(ω) is unknown to start with, so anarbitrary initial phase is assumed. The choice of the initial phaseφ₀(ω) does not significantly affect the convergence of the method 400.Therefore, in certain embodiments, the initial phase φ₀ (ω) isconveniently chosen to be equal to zero.

The method 400 further comprises multiplying the measured transmissionspectrum amplitude |t_(M)(ω)| and the estimated phase term exp[jφ₀(ω)]to generate an estimated complex transmission spectrum t′(ω) in anoperational block 430. The method 400 further comprises calculating anintermediate function h′(t), which is the inverse Fourier transform(IFT) of the estimated complex transmission spectrum t′(ω), in anoperational block 440. The method 400 further comprises calculating anew estimated complex transmission impulse response of the firstiteration h₁(t) by applying at least one constraint to the IFT of theestimated complex transmission spectrum, in the operational block 450.For example, in certain embodiments, since the IFT of t(ω) is known tobe causal, only the t≧0 portion of h′(t) is retained and zeros are usedfor t<0, thereby providing a new estimated complex transmission impulseresponse of the first iteration h₁(t). As described above, various othercharacteristics of the complex transmission spectrum t′(ω) may be usedas sources for the applied constraint, which include but are not limitedto, finite temporal duration, finite bandwidth of the FBG, uniformity ofthe FBG, and using only the real portion of the complex transmissionspectrum t′(ω).

The method 400 further comprises calculating a Fourier transform t₁(ω)of the estimated complex transmission impulse response h₁(t) in anoperational block 460. The method 400 further comprises calculating aphase term exp[jφ₁(ω)] of the Fourier transform t₁(ω) of the estimatedcomplex transmission impulse response h₁(t) in an operational block 470.The method 400 further comprises using the calculated phase tennexp[jφ₁(ω)] as the estimated phase term in the operational block 430 andrepeating the operational blocks 430, 440, 450, 460, and 470, as denotedby the arrow 480. In certain such embodiments, the calculated phase termexp[jφ_(n)(ω)] of an interation n provides a new estimate for themissing phase term of the complex transmission spectrum. The resultingestimated Fourier transform of the operational block 430 is the productof a measured transmission spectrum amplitude |t_(M)(ω)| and acalculated estimated phase term exp[jφ_(n)(ω)]. By repeating theoperational blocks 430, 440, 450, 460, and 470, a second estimatedcomplex transmission impulse response and a second estimated phase termare generated.

In certain embodiments, the operational blocks 430, 440, 450, 460, 470are iteratively repeated a number of times, as shown in FIGS. 4A and 4B.In certain such embodiments, the iterations are performed until theresulting estimated complex transmission impulse response h_(n)(t)converges. Convergence is reached in certain embodiments when theaverage difference between the estimated complex transmission impulseresponses obtained after two consecutive iterations (e.g.,|h_(n)(t)−h_(n-1)(t)|²/h_(n)(t)|²) is less than a predetermined value(e.g., 0.1% of the estimated complex transmission impulse response ofthe iteration). In other embodiments, the iterations are performed apredetermined number of times (e.g., 100 times) rather than determiningthe differences between successive iterations. In certain embodiments,the predetermined number is selected to be sufficiently large such thatconvergence is essentially always achieved after this number ofiterations. In certain such embodiments, the predetermined number isdetermined by evaluating the rate of convergence for a number of FBGs.At the end of the n^(th) iteration, the phase of the Fourier transformof h_(n)(t) is the recovered phase φ_(n)(ω) of the FBG complextransmission spectrum t(ω). In certain embodiments, the transmissiongroup delay dφ_(n)(ω)/dω is then calculated.

In certain embodiments, the phase φ_(n)(ω) obtained by the method 400converges to the minimum-phase function (MPF) corresponding to a givenFourier transform amplitude. MPFs have the property that the FT phaseand the logarith of the FT amplitude of an MPF are the Hilbert transformof one another. Consequently, the FT phase of an MPF can always berecovered from its FT amplitude, and an MPF can always be reconstructedfrom its FT amplitude alone. (See, e.g., V. Oppenheim and R. W. Schafer,“Digital Signal Processing,” 2000, Prentice Hall, Chapter 7.) The factthat the complex transmission spectrum of any FBG is an MPF, asdescribed more fully below, ensures the convergence of the method 400 tothe unique phase φ_(t)(ω) of the complex transmission spectrum, i.e.,φ_(n) (ω)=φ_(t)(ω).

Since the IFT of the complex reflection spectrum r(ω) is not generally aminimum-phase function, the method 400 is not generally applicable torecover φ_(r)(ω) from only the measured reflection spectrum amplitude|r_(M)(ω)|. However, for certain embodiments in which the complexreflection spectrum r(ω) is also a minimum-phase function (e.g., foruniform FBGs), the method 400 can be used to derive the reflection groupdelay, as well as the transmission group delay, of the FBG.

In certain embodiments in which the FBG is known to be symmetric, thereflection group delay of the FBG is equal to the transmission groupdelay. In certain such embodiments, both the reflection group delay andthe transmission group delay are recovered using the method 400.

The method 400 has been used to recover the transmission group delayspectra of various exemplary FBGs to illustrate the usefulness of themethod 400. FIGS. 5A-5C and 6A-6C schematically display two suchexamples. FIGS. 5A-5C schematically illustrate the results for asymmetric Gaussian-apodized FBG. FIGS. 6A-6C schematically illustratethe results for a non-symmetric chirped FBG. The amplitudes of thereflection spectra of both FBGs are shown in FIG. 5A and FIG. 6A,respectively. In FIG. 5B and FIG. 6B, the theoretical transmission groupdelays are shown with the solid curves, whereas the recovered groupdelays (without any noise present) recovered using the method 400 (withn=100 iterations) are shown with the dashed curves. These computationseach took less then a few seconds with a 500-MHz computer. The recoveryfor each FBG is so good for each example that it is difficult todistinguish the theoretical group delay curve (solid line) from therecovered group delay curve (dashed line).

To demonstrate the noise performance of the method 400, the theoretical|t(ω)|² spectrum was multiplied by 5% uniform noise with a mean of 1,and the method 400 was then applied to this noisy transmission spectrum.The results of the group delay recovery, with again n=100 iterations,are shown in FIG. 5C and FIG. 6C by dotted curves. Once again therecovery is very good, showing that the method 400 can even be quiteuseful under severe noise.

The vertical scale in FIG. 6B is enlarged so that the small dc offset(on the order of approximately 0.1 to 0.2 picoseconds) between thetheoretical and the recovered group delay curves is more visible. Withother techniques (e.g., the technique of Poladian referenced above),much stronger dc offsets, on the order of approximately 20 to 30picoseconds, are observed. The main cause of this small dc offset forthe method 400 is the usage of limited bandwidth in the simulations. Forexample, if a wider wavelength window (greater than 4 nanometers) wereused for FIG. 6A, the recovery results would have been much moreimproved.

Iterative Processing Utilizing Spectral Interferometry

Certain embodiments described herein use a pulsed input laser and an OSAto retrieve both the amplitude and the phase of the reflection ortransmission spectrum of an FBG using a single power spectrummeasurement. Certain embodiments are referred to herein as spectralinterferometry using minimum-phase based algorithms (SIMBA).

FIGS. 7A and 7B schematically illustrate two exemplary measurementconfigurations compatible with embodiments described herein. While thesemeasurement configurations are slightly different, the processing of themeasured quantities for the characterization of the FBG, together withthe principle of operation, is identical.

In certain embodiments, the measurement configuration 500 of FIG. 7A hasa pulsed laser source 510 which generates a laser pulse 520 which issplit by a beam splitter 530 with a preferably uneven splitting ratio R,where R>1, into a first laser pulse 522 and a second laser pulse 524.The second laser pulse 524 is then sent to the FBG 540 beingcharacterized. In certain embodiments, the pulsed laser source 510comprises a mode-locked laser with a temporal width of a few picoseconds(e.g., approximately 2 picoseconds to approximately 4 picoseconds). Incertain embodiments, the splitting ratio of the beam splitter 530 isbetween approximately 1 and approximately 200, while other embodimentshave even higher splitting ratios.

In certain embodiments, the reflected pulse 550 from the FBG 540 iscollected using circulators 575, 576 and temporally combined with thedelayed first laser pulse 522, thereby forming a pulse sequence 560. Thepulse sequence 560 has a sharp peak at the leading edge, due to thefirst laser pulse 522, followed by a broader and much weaker pulse whichis due to the reflected pulse 550 from the FBG 540. The pulse sequence560 is then sent to an OSA 570, which yields the power spectrum or thesquare of the Fourier transform (FT) amplitude of the electric fieldenvelope of the pulse sequence 560. As is described more fully below,the pulse sequence 560 inputted into the OSA 570, which has a sharp peakat its leading edge, is close to a minimum-phase function (MPF), whichmakes recovery of its FT phase possible from only the measurement of itsFT amplitude, or vice versa. This recovery is performed by processingthe measured power spectrum either analytically or iteratively, whichyields both the phase and the amplitude of the reflection ortransmission spectrum of the FBG 540.

In certain embodiments, the optical path length between the beamsplitter 530 and the FBG 540, the optical pathlength between the beamsplitter 530 and the OSA 570, and the optical pathlength between the FBG540 and the OSA 570 are selected to provide a predetermined time delay rbetween the portions of the pulse sequence 560 received by the OSA 570(e.g., the first laser pulse 522 and the reflected pulse 550 from theFBG 540). In certain embodiments, at least one of the optical pathsbetween the beam splitter 530, the FBG 540, and the OSA 570 includes adelay line which can be adjusted to provide a desired time delay T inthe pulse sequence 560.

In certain other embodiments, the beam splitter 530 has a splittingratio of approximately one, and an attenuator is placed between the beamsplitter 530 and the circulator 575. In certain other embodiments, abeam splitter 530 having a splitting ratio greater than one and anattenuator between the beam splitter 530 and the circulator 575 areused. In certain embodiments, the splitting ratio of the beam splitter530 is adjustable. In certain embodiments, the attenuation of theattenuator is adjustable.

In certain other embodiments, a measurement configuration 580, such asthat schematically illustrated by FIG. 7B, utilizes a coupler 581, anattenuator 582, and a mirror 584. In certain embodiments, the splittingratio of the coupler 581 is approximately equal to one, the mirror 584is a high reflector, and the attenuator 582 is adjusted such that theamplitude of the reflected spectrum from the FBG 540 is much smallerthan the amplitude of the reflected laser pulse from the mirror 584. Incertain such embodiments, the reflected laser pulse from the mirror 584and the reflected spectrum from the FBG 540 are combined by the coupler581 to form a function which approximates a minimum-phase function.

In certain embodiments, the optical path length between the laser source510 and the FBG 540, the optical pathlength between the laser source 510and the mirror 584, the optical pathlength between the FBG 540 and theOSA 570, and the optical pathlength between the mirror 584 and the OSA570 are selected to provide a predetermined time delay τ between theportions of the pulse sequence 560 received by the OSA 570. In certainembodiments, at least one of the optical paths between the laser source510, the FBG 540, the mirror 584, and the OSA 570 includes a delay linewhich can be adjusted to provide a desired time delay τ in the pulsesequence 560.

In certain other embodiments, the coupler 581 has a splitting ratiogreater than one, and the attenuator 582 between the coupler 581 and theFBG 540 is removed. In certain other embodiments, both a coupler 581having a splitting ratio greater than one and an attenuator 582 betweenthe coupler 581 and the FBG 540 are used. In certain embodiments, thesplitting ratio of the coupler 581 is adjustable. In certainembodiments, the attenuation of the attenuator 582 is adjustable.

Certain embodiments provide advantages over previously-existingtechniques, particularly over low-coherence spectral interferometry. Incertain embodiments, the time delay r between the reflected pulse 550and the first laser pulse 522 can be chosen arbitrarily small as long asthe two pulses temporally do not overlap. In certain embodiments, thisis especially important if the OSA 570 used for the power spectrummeasurement does not have enough resolution to resolve the spectralinterference fringes, since the larger the time delay τ between the twopulses, the higher the maximum frequency of oscillations in the powerspectrum recorded by the OSA 570. Certain embodiments advantageouslyallow characterization of more than one FBG at the same time by using asingle OSA measurement. Certain embodiments advantageously utilize alaser pulse 520 with a temporal profile width significantly narrowerthan the impulse response of the FBG 540 being characterized. Incontrast, for low-coherence spectral interferometry, the autocorrelationfunction of the pulsed laser source is required to be much narrower thanthe impulse response of the FBG. In other words, for the same FBG,certain embodiments described herein give roughly two times betterresolution than does low-coherence spectral interferometry.

FIG. 8 schematically illustrates another exemplary measurementconfiguration 600 compatible with certain embodiments described herein.A pulsed laser source 610 generates an input laser pulse 620 having atemporal width (e.g., 50 picoseconds). The input laser pulse 620 issplit into a first laser pulse 622 which is directed to a delay line 630and a second laser pulse which is directed to a pulse stretcher 640(e.g., a loop of fiber optic cable). The pulse stretcher 640 broadensthe temporal width of the second laser pulse by a predetermined factor(e.g., approximately 2 to 5) to produce a stretched laser pulse 624. Thestretched laser pulse 624 is sent through an attenuator 650, and is thensplit into two weak pulses. One of the weak pulses is directed to theFBG 540 being characterized, while the other weak pulse is reflectedfrom a mirror 660 (e.g., a bare fiber end, a mirrored fiber end, or amirror placed at the end of the fiber with a collimating lenstherebetween). The reflected pulse 672 from the FBG 540 and thereflected pulse 674 from the mirror 660 are temporally combined with thetime-delayed version of the first laser pulse 622. The resultant pulsesequence 680 has a dominant pulse at the leading edge followed by twoweaker reflected pulses and is received by the OSA 570, which measuresthe power spectrum. With proper selection of the relative amplitudes ofthe pulses forming the pulse sequence 680 (e.g., amplitude of the laserpulse 622 is significantly larger than the maximum amplitude of thereflected pulses 672, 674), the pulse sequence 680 approximates aminimum-phase function. In certain such embodiments, the pulse sequence680 is processed iteratively to recover the reflected pulse 672 from theFBG 540. The processing of the measured FT amplitude from the OSA 570,together with the recovery algorithm described more fully below, are thesame as for the measurement configurations 500, 580 of FIGS. 7A and 7B.

Certain embodiments of the measurement configuration 600 schematicallyillustrated by FIG. 8 provide the same advantages as described above inrelation to the measurement configurations 500, 580 of FIGS. 7A and 7B.In certain embodiments, the measurement configuration 600 of FIG. 8provides other advantages as well. In certain embodiments, themeasurement configuration 600 advantageously does not require that theinput laser pulse be temporally much narrower than the impulse responseof the FBG 540. For example, low-coherence spectral interferometrytypically requires laser pulse widths of approximately 1 picosecond toaccurately characterize an FBG with an impulse response of approximately100 picoseconds duration. In contrast, certain embodiments utilizing themeasurement configuration 600 advantageously avoid this constraint onthe laser pulse width. For example, laser pulse widths of 50 picosecondscould be used to fully characterize the impulse response of the FBG 540.

Mathematically, the reflection impulse response h_(R)(t) and thetransmission impulse response h_(T)(t) of an FBG are simply the inverseFourier transforms of the complex reflection spectrum r(ω) and thecomplex transmission spectrum t(ω), respectively. Therefore, knowing theimpulse response of an FBG also means knowing the complex spectrum ofthe FBG. Since the impulse response of an FBG is a complex quantity,measuring the impulse response of an FBG is as equally challenging asmeasuring the whole complex spectrum of the FBG.

The transmission impulse response, h_(T)(t), where t is the relativetime, of all FBGs belong to the class of minimum-phase functions (MPFs).(See, e.g., L. Poladian, “Group-Delay Reconstruction for Fiber BraggGratings in Reflection and Transmission,” Optics Letters, 1997, Vol. 22,pp. 1571-1573; J. Skaar, “Synthesis of Fiber Bragg Gratings for Use inTransmission,” J. Op. Soc. Am. A, 2001, Vol. 18, pp. 557-564.) An MPF ischaracterized by having a z-transform with all its poles and zeros on orinside the unit circle. As a result of this property, the FT phase andthe logarithm of the FT amplitude of an MPF are the Hilbert transform ofone another. Consequently, the FT phase of an MPF can always berecovered from its FT amplitude, and an MPF can always be reconstructedfrom its FT amplitude alone. This property of MPFs is used in certainembodiments described herein to retrieve the whole complex transmissionspectrum t(ω) from only the measured reflection spectrum amplitude|r(ω)| or the measured transmission spectrum amplitude |t(ω)| (since fora lossless grating |t(ω)|²+|r(ω)|²=1). This recovery of the complexreflection spectrum or the complex transmission spectrum can be achievedby either analytical or iterative techniques. (See, e.g., A. Ozcan etal., “Group Delay Recovery Using Iterative Processing of Amplitude ofTransmission Spectra of Fibre Bragg Gratings,” Electronics Letters,2004, Vol. 40, pp. 1104-1106.) Generally, the reflection impulseresponse, h_(R)(t) of an FBG is not an MPF, so the whole complexreflection spectrum r(ω) cannot be uniquely recover from only theamplitude spectrum measurement. However, once the complex reflectionspectrum r(ω) has been characterized by some means, the complextransmission spectrum t(ω) is also fully characterized using|t(ω)|²=1−|r(ω)|², since t(ω) can fully be recovered from only |t(ω)|due to the above mentioned MPF property. Certain embodiments describedherein are the first application of the concept of MPFs to thecharacterization of FBG spectra, either reflection or transmission.

FIG. 9 is a block diagram of a method 700 of characterizing an FBG inaccordance with certain embodiments described herein. The method 700includes an iterative error-reduction algorithm which uses the measuredFT amplitude |E_(M)(ƒ)| of an unknown function e(t), together with theknown properties of e(t) (e.g., being causal), and iteratively correctsan initial guess for e(t). (See, e.g., J. R. Fienup, “Reconstruction ofan Object from the Modulus of its Fourier Transform,” Optics Letters,1978, Vol. 3, pp. 27-29; R. W. Gerchberg and W. O. Saxton, “PracticalAlgorithm for the Determination of Phase from Image and DiffractionPlane Pictures,” Optik, 1972, Vol. 35, pp. 237-246.)

In certain embodiments, the method 700 recovers the complex electricfield envelope e(t) of the impulse response of an FBG. In certainembodiments, the OSA output 710 is used to provide the measured FTamplitude |E_(M)(ƒ)| of a complex MPF, e(t). As shown by the block 720,e(t) is the only quantity inputted into the method 700. In certainembodiments, the function e(t) is an exact MPF, while in certain otherembodiments, the function e(t) only approximates an MPF. (See, e.g., T.F. Quatieri, Jr. and A. V. Oppenheim, “Iterative Techniques for MinimumPhase Signal Reconstruction from Phase or Magnitude,” IEEE Trans. onAcoustics, Speech, and Signal Processing, 1981, Vol. 29, pp. 1187-1193;M. Hayes et al., “Signal Reconstruction from Phase or Magnitude,” IEEETrans. Acoustics, Speech, and Signal Processing, 1980, Vol. 28, pp.672-680; A. Ozcan et al., “Iterative Processing of Second-Order OpticalNonlinearity Depth Profiles,” Optics Express, 2004, Vol. 12, pp.3367-3376.)

Prior to the method 700 being performed, the FT phase spectrum isunknown. Therefore, in certain embodiments, an arbitrary initial guessφ₀(ƒ) is used for the phase spectrum, as shown by the block 730.Generally, this initial guess of the phase does not significantly affectthe accuracy of the result of the method 700. For this reason, incertain embodiments, φ₀(ƒ) is conveniently chosen to be equal to zero.

Using the measured FT amplitude |E_(M)(ƒ)| and the initial guess of thephase φ₀(ƒ), the inverse Fourier transform (IFT)e′(t)=|E_(M)(ƒ)|·exp(jφ₀) is calculated numerically, as shown by theblock 740. The method 700 further comprises applying at least oneconstraint to the IFT, in the operational block 750. For example, incertain embodiments, since all MPFs are causal, only the t≧0 portion ofe′(t) is retained, while all values of e′(t) for t<0 are set to zero, asindicated by the block 750 of FIG. 9. In certain embodiments in whiche(t) is known to be limited in time (e.g., finite temporal duration ofless than 100 picoseconds), in the calculation of block 750, the valuesof e′(t) for t>100 picoseconds are also set to zero, which speeds upconvergence. As described above, various other characteristics of e′(t)may be used as sources for the applied constraint, which include but arenot limited to, finite bandwidth of the FBG, uniformity of the FBG, andusing only the real portion of e′(t).

In certain embodiments, the result of the calculation of the block 750is a new function e₁(t), which is the first estimate of the complex MPFe(t). In certain embodiments, the FT of the first estimate e₁(t) iscalculated, as indicated by the block 760, thereby providing a new phaseφ₁(f) and a new amplitude |E₁(ƒ)| for the FT of e(t).

Since the amplitude of the FT spectrum must be equal to the measuredamplitude |E_(M)(ƒ)|, |E₁(ƒ)| is replaced by |E_(M)(ƒ)| and the loop isrepeated using |E_(M)(ƒ)| and φ₁(ƒ) as the new input spectra to theblock 740, which provides a second function e₂(t). This loop is repeatedn times, until convergence is achieved. In certain embodiments,convergence is achieved once the difference between two consecutiveestimates of the function ∫|e_(n)(t)−e_(n-1)(t)|² dt/∫|e_(n)(t)|² isless than a predetermined value (e.g., 0.1%). At the end of the n-thiteration, e_(n)(t) is the recovered complex MPF, as indicated by theblock 770. Typically, approximately 100 iterations are adequate forachieving convergence, which only takes a few seconds to compute usingMATLAB on a 500 MHz computer with 2¹⁴ data points.

Although it has not been proven mathematically, it has been foundempirically that the method 700 always converges to the minimum-phasefunction corresponding to a given FT amplitude. In other words, of theinfinite family of FT phase functions that can be associated with theknown (measured) FT amplitude, the method 700 converges to the one andonly one that has the minimum phase. Since this solution is unique, ifthe profile to be reconstructed is known a priori to be an MPF or toapproximate an MPF, then the solution provided by the error-reductionmethod 700 is the correct profile.

To understand intuitively which physical functions are likely to beminimum-phase functions, an MPF is denoted by e_(min)(n), where n is aninteger that labels sampled values of the function variable, e.g.,relative time in the present case of ultra-short pulses. Because allphysical MPFs are causal, e_(min)(n) must be equal to zero for n<0. Theenergy of a minimum-phase function, which is defined as$\sum\limits_{n = 0}^{m - 1}\quad{{e_{\min}(n)}}^{2}$for m samples of the function, satisfies the inequality${\sum\limits_{n = 0}^{m - 1}\quad{{e_{\min}(n)}}^{2}} \geq {\sum\limits_{n = 0}^{m - 1}\quad{{e(n)}}^{2}}$for all possible values of m>0. In this inequality, e(n) represents anyof the functions that have the same FT amplitude as e_(min)(n). Thisproperty suggests that most of the energy of e_(min)(n) is concentratedaround n=0. Stated differently, any profile with a dominant peak aroundn=0 (e.g., close to the origin) will be either a minimum-phase functionor close to one, and thus it will work extremely well with the iterativeerror-reduction method 700 schematically illustrated by FIG. 9. Althoughthere might be other types of MPFs besides functions with a dominantpeak, this class of MPFs is advantageously used because they arestraightforward to engineer with optical pulses and because they yieldexceedingly good results.

In certain embodiments, the method 700 can be used to uniquelycharacterize the whole complex reflection and hence transmission spectraof any FBG by recovering the reflection impulse response, h_(R)(t),using a single FT amplitude measurement. Certain embodiments describedherein rely on the fact that by increasing the peak amplitude of theleading pulse in a sequence of pulses, the entire pulse sequence (evenif the sequence is a complex quantity, such as the electric fieldenvelope) becomes close to an MPF, which makes the recovery of the phaseinformation possible from only the FT amplitude measurements.

Referring to the measurement configurations of FIGS. 7A and 7B, a shortlaser pulse 524 with a complex electric field envelope of E(t) impingesan FBG 540 being characterized. The reflected pulse spectrum 550 cansimply be calculated as E_(Ref1)(ω)=E(ω)·r(ω), where E(ω) is simply theFT of E(t). (See, e.g., L. R. Chen et al., “Ultrashort Pulse ReflectionFrom Fiber Gratings: A Numerical Investigation,” J. of LightwaveTechnology, 1997, Vol. 15, pp. 1503-1512.) For simplicity andconvenience, the term exp(j·ω_(c)), where ω_(c) corresponds to thecenter frequency of the input laser 510 has been dropped. In the timedomain, the reflected pulse envelope 550 can then be expressed asE_(Ref1)(t)=E(t)*h_(R)(t), where ‘*’ stands for the convolutionoperation. Therefore, the complex electric field envelope of thereflected pulse 550 is simply a convolution of the reflection impulseresponse of the FBG 540 with the complex electric field of the inputpulse 524. For embodiments in which the input laser pulse 520 istemporally much narrower than the impulse response of the FBG 540, thereflected pulse 550 from the FBG 540 E_(Ref1)(t) approximates or equalsthe reflection impulse response h_(R)(t). Most commercially availableFBGs operate at a wavelength of approximately 1550 nanometers, so thetemporal width of the reflection impulse response of such an FBG isapproximately 50 picoseconds to approximately 100 picoseconds, orlonger. Thus, in certain embodiments, an input laser pulse 520 having atemporal width of a few picoseconds practically acts as a delta functionto yield E_(Ref1)(t)=h_(R)(t). However, in terms of measurementcomplexity, since both E_(Ref1)(t) and h_(R)(t) are complex quantities,to recover these complex quantities, two independent measurements wouldtypically be used, one for the amplitude and the other for the phase.

Referring to the measurement configuration 500 of FIG. 7A, the reflectedimpulse response h_(R)(t) of the FBG 540 is combined in the time domainwith the second laser pulse 522, denoted by E_(A)(t), with a time delayof τ between the two pulses. The resulting pulse sequence 560, expressedas E_(Seq)(t)=E_(A)(t)+h_(R)(t−τ), is then sent to the OSA 570, whichyields the optical power spectrum or the square of the FT amplitude ofthe complex electric field envelope of the pulse sequence 560, expressedas |E_(Seq)(ω)|², where E_(Seq)(ω) is the FT of E_(Seq)(t). Using onlythe measured FT amplitude square |E_(Seq)(ω)|², certain embodimentsrecover the reflection impulse response h_(R)(t) based on the propertiesof MPFs. In certain embodiments, E_(Seq)(t) is selected to approximate atrue MPF by increasing the peak amplitude of E_(A)(t), and recoveringits FT phase from only its FT amplitude |E_(Seq)(ω)|. In certainembodiments, the recovery of the FT phase spectrum is achieved by usingan analytical Hilbert transformation. However, in certain otherembodiments, an iterative error-reduction method, such as the method 700schematically illustrated in FIG. 9, is preferably used to recover thephase spectrum, due to the simplicity and better noise performance ofthe method.

A numerical example illustrates the recovery of E_(Ref1)(t)=h_(R)(t)from only the measured quantity, |E_(Seq)(ω)|² or |E_(Seq)(ω)| using themeasurement configuration of FIG. 7A and the method 700 of FIG. 9. Astrongly chirped asymmetric FBG is used in this numerical example tomake the recovery a harder task. The theoretical electric fieldreflection coefficient amplitude and phase of the chosen chirped FBG,which was calculated using a transfer matrix formalism (see, e.g., T.Erdogan, “Fiber Grating Spectra,” J. of Lightwave Technology, 1997, Vol.15, pp. 1277-1294), are shown in FIGS. 10A and 10B, respectively. Thereflection band of this FBG is approximately 4 nanometers wide, betweenapproximately 1548 nanometers and approximately 1552 nanometers. Theamplitude of the theoretical reflection impulse response of this FBG isshown in FIG. 11. The temporal width of the reflection impulse responseis approximately 300 picoseconds. The observed broadened temporalbehavior is mostly due to the strong chirp of the reflectioncoefficient. As expected, the reflection impulse response of the FBGshown in FIG. 11 is causal.

FIGS. 12A and 12B illustrate the amplitude and phase, respectively, ofthe temporal profile of the input laser pulse used in the exemplarynumerical simulation of FIGS. 10A, 10B, and 11. The temporal width ofthe chosen laser pulse is less than approximately 6 picoseconds, suchthat the laser pulse approximates as a delta function for the reflectionimpulse response of FIG. 11. The normalized amplitude of the input laserpulse spectrum together with the reflection spectrum of the exemplaryFBG of FIGS. 10A and 10B are also plotted in FIG. 13. In certainembodiments, not only does the input laser pulse have a temporal widthmuch narrower than the impulse response of the FBG, whereby the inputlaser pulse acts as a delta function, but also the input laser pulse hasa power spectrum that covers all the frequencies in the FBG reflectionspectrum, as shown in FIG. 13.

Using the measurement configuration of FIG. 7A, the amplitude (solidline) and the phase (dashed line) of the pulse sequence 560 is formed,as plotted in FIG. 14. The pulse sequence 560 includes the laser pulse522 and the weaker reflected pulse 550, which approximates thereflection impulse response of the FBG 540. A splitting ratio of 40 wasused between the peak amplitude of the laser pulse 522 and the peakamplitude of the reflection impulse response 550 of the FBG. In certainembodiments, this splitting ratio of 40 ensures that the complexelectric field envelope of the pulse sequence is close to an MPF touniquely recover h_(R)(t) from only the measurement of the powerspectrum of the pulse sequence.

The pulse sequence 560 is then sent to the OSA 570, which records thepower spectrum or the square of the FT amplitude of complex electricfield envelope of the pulse sequence, as shown in FIG. 14. Thecalculated power spectrum is plotted in FIGS. 15A and 15B for asplitting ratio of 40. To calculate the power spectra of FIGS. 15A and15B, the resolution of the OSA 570 was assumed to be limited byapproximately 10 picometers, which is a modest resolution for currentlyavailable spectrum analyzers. Other OSAs compatible with embodimentsdescribed herein have sub-picometer resolution. FIG. 15B shows anenlarged view of the same power spectrum of FIG. 15A in the range ofwavelengths between approximately 1550 nanometers and approximately 1551nanometers. FIG. 15B illustrates that due to the limiting 10 picometerresolution of the OSA 570, some sharp features of the power spectrumcurve are actually lost. As discussed more fully below, the wholecomplex spectrum of the target FBG can still be recovered accurately.The fringe pattern shown in FIGS. 15A and 15B is the result ofinterference between the input laser pulse spectra and the FBGreflection spectra. This interference is only observed in the frequencyband of the reflection spectrum of the FBG, and the overall envelope ofthe power spectrum in FIGS. 15A and 15B follows the power spectrum ofthe input laser pulse.

FIGS. 16A and 16B show the amplitude spectrum and phase spectrum,respectively, of the complex reflection impulse response of the targetFBG recovered by inputting the output of the OSA 570 into the iterativeerror-reduction method 700 shown in FIG. 9. The solid and dashed curvesof FIG. 16A correspond to the original and recovered amplitudes,respectively, of the impulse response. The solid and dashed curves ofFIG. 16B correspond to the original and recovered phases, respectively,of the impulse response. The recovered impulse response is an excellentfit to the original impulse response. In fact, it is difficult to seeany difference between the solid and dashed curves of FIG. 16A, and thedifference between the solid and dashed curves of FIG. 16B are small.

The time origin information for the recovered reflection impulseresponse is lost in FIGS. 16A and 16B. In principle, the origin can beredefined by using the causality property of the impulse response, i.e.,by choosing the point to the left of which the recovered impulseresponse is all zero, as the new time origin. Any error in theredefinition of the time origin is, however, inconsequential, since atime shift in the impulse response merely adds a linear phase to its FTphase. The recovered group delay will therefore only have a constantoffset, which will be proportional to the error made in the time origin.This constant group delay offset is inconsequential for practicalapplications and can simply be recovered by noting the value of thegroup delay away from the FBG reflection spectrum.

FIGS. 16A and 16B do not show the recovery of the leading input pulseelectric field since it is not of interest. The input laser pulse simplyacts as a dummy pulse for the recovery. The data processing does recoverthe leading input pulse if FIG. 14 as well, but it does not generallyrecover it accurately, primarily because the pulse sequence onlyapproximates a true MPF. In other words, in certain embodiments in whichthe pulse sequence only approximates a true MPF, the recovery of theleading input laser pulse's electric field is affected, but the accuracyof the recovered reflection impulse response of the FBG is not affected.However, any errors in the recovered input laser pulse profile areinconsequential since the input laser pulse profile is not the target ofthe analysis. In certain embodiments, the temporal profile of the inputlaser pulse is selected or engineered to result in a pulse sequencewhich is a more exact approximation of an MPF. Certain such embodimentsadvantageously increase the speed of convergence of the calculations orallow the application of the Hilbert transformation in the calculations.

From the recovered impulse response of the FBG, shown in FIGS. 16A and16B, the reflection coefficient amplitude and the group delay spectra ofthe FBG are easily computed by a single FT operation, as shown in FIGS.17A and 17B. The solid and dashed curves of FIG. 17A correspond to theoriginal and recovered amplitudes, respectively, of the reflectioncoefficient. The solid and dashed curves of FIG. 17B correspond to theoriginal and recovered group delay spectra, respectively. Even for astrongly chirped FBG, as shown in FIGS. 10A and 10B, the success in therecovery is quite impressive. For this exemplary embodiment, the errorin the recovery of |r(ω)|, defined as$\frac{\int{{{{{r(\omega)}}\quad - \quad{\quad{\hat{r}(\omega)}}}}^{2}{\mathbb{d}\omega}}}{\int{{{r(\omega)}}^{2}{\mathbb{d}\omega}}},$where |r(ω)| and |{circumflex over (r)}(ω)| are the original and therecovered quantities, respectively, is only approximately 0.08%.

To evaluate the dependence of the success of the recovery on thetemporal profile of the chosen input laser pulse, another exemplaryembodiment uses a different input laser pulse having an amplitude andphase as shown in FIGS. 18A and 18B, respectively. The laser pulse ofFIGS. 18A and 18B is approximately 3 times narrower than the input laserpulse of the previous exemplary embodiment, as seen by comparing FIGS.12A and 12B with FIGS. 18A and 18B. For this exemplary embodiment, asplitting ratio of 120 was used between the peak amplitude of the inputlaser pulse and the peak amplitude of the reflection impulse response ofthe FBG in FIG. 7A.

FIGS. 19A and 19B show the amplitude and group delay spectra,respectively, of the reflection coefficient of the FBG. The solid anddashed curves of FIG. 19A are the original and recovered amplitudes,respectively, of the reflection coefficient. The solid and dashed curvesof FIG. 19B are the original and recovered group delay spectra,respectively. The error in the recovery of |r(ω)| in this exemplaryembodiment is less than 0.02%, which is reduced by more than a factor of4 as compared to the error of the previous exemplary embodiment. Theimproved performance in this exemplary embodiment (0.02% versus 0.08% inthe previous example) is primarily due to the narrower input laserpulse, which more closely approximates a true delta function, therebyyielding a more accurate reflected pulse that represents the truereflection impulse response of the FBG.

FIG. 20 is a plot of the results from a series of simulations, each witha different splitting ratio between the peak amplitude of the inputlaser pulse and the peak amplitude of the reflected impulse response.The two curves of FIG. 20 correspond to the two input laser pulses shownin FIGS. 12A and 12B (“longer laser pulse”) and FIGS. 18A and 18B(“shorter laser pulse”). FIG. 20 shows the logarithm of the error in therecovery of |r(ω)| as a function of the logarithm of the splittingratio.

As shown by FIG. 20, the error in the recovery generally decreases forincreasing splitting ratios between the peak amplitude of the inputlaser pulse and the peak amplitude of the reflected impulse response ofthe FBG. For embodiments with splitting ratios below a splitting ratioof approximately 100, the longer laser pulse has a smaller error in therecovery than does the smaller laser pulse. However, for splittingratios larger than a critical ratio of approximately 100, using ashorter input pulse provides a much better recovery than does using alonger input laser pulse (e.g., recovery improved by a factor ofapproximately 4). In addition, while the baseline error for the longerlaser pulse is achieved at lower splitting ratios (e.g., greater thanapproximately 26), it is higher than the baseline error achieved usingthe shorter laser pulse, even though the shorter laser pulse's baselineerror is only achieved with higher splitting ratios (e.g., greater thanapproximately 100).

The reason for this observed lower baseline in error for the shorterlaser pulse is that the shorter input laser pulse more closelyapproximates a true delta function, thereby yielding a more accuratereflected pulse to represent the true reflection impulse response of theFBG and reducing the error in the recovery. However, by narrowing theinput laser pulse, the energy of the input laser pulse (proportional tothe area under the temporal profile of the laser pulse) is also reduced.Therefore, higher splitting ratios are used to obtain the benefits ofthe reduced error from a more narrow input laser pulse. In other words,when using a shorter or narrower dummy input pulse, it is advantageousto use a stronger peak amplitude (or a higher critical ratio) for theinput pulse. In certain embodiments, as shown by FIG. 20, the error isdecreased by a factor of approximately 4 by using an input laser pulsethat is temporally approximately 3 times narrower.

In certain embodiments, the ratio of the integral of the normalizedlaser field of the input laser pulse to the integral of the normalizedreflection impulse response of the FBG, i.e.,$\int{\frac{{E_{Pulse}(t)}}{\max\left( {{E_{Pulse}(t)}} \right)}{{\mathbb{d}t}/{\int{\frac{{h_{R}(t)}}{\max\left( {{h_{R}(t)}} \right)}{\mathbb{d}t}}}}}$is useful to select an optimum temporal width of the input laser pulseand the splitting ratio. For the shorter laser pulse of FIG. 20, thisratio is approximately 0.35%, and for the longer laser pulse, this ratiois approximately 1.4%. In other words, for the shorter laser pulse, thetotal area under the normalized electric field amplitude of the inputlaser pulse is only 0.35% of the total area under the normalizedamplitude of the reflection impulse response of the target FBG. This lowratio is expected since the input laser pulse approximates a deltafunction. However, this ratio increases to 1.4% for the longer pulse(e.g., by approximately a factor of 4), while the critical splittingratio drops by a factor of approximately 4 as well (e.g., from 100 to26). Thus, for broader impulse response FBGs (such as strongly chirpedFBGs), a generally larger critical ratio is advantageously used toachieve convergence. This behavior can also be related to the propertyof MPFs that most of the energy of an MPF is concentrated in proximityto the origin, as discussed above. In certain embodiments, for a broaderreflection impulse response of the FBG, a higher peak amplitude of theleading dummy pulse is advantageously used to satisfy this property ofMPFs for the input pulse sequence.

FIG. 21 is a plot of a simulated noisy power spectrum used to illustratehow measurement errors in the power spectrum affect the accuracy of therecovery results. The power spectrum of FIG. 21 was produced bymultiplying the theoretical FT amplitude square of an input pulsesequence by a uniform random noise (e.g., 10% peak-to-peak amplitudewith an average of unity). The error-reduction method 700 of FIG. 9 wasapplied to the simulated noisy power spectrum of FIG. 21 to recover thereflection impulse response of the target FBG. In this exemplaryembodiment, the resolution of the OSA was assumed to be approximately 10picometers and a splitting ratio of 27 between the optical fields wasused (corresponding to a splitting ratio of 27² of the optical powers).

The calculated reflection coefficient amplitude and the group delayspectra of the target FBG, computed from the measured power spectrum ofFIG. 21, are shown in FIGS. 22A and 22B. The recovery is still quitegood despite the strong noise added to the power spectrum used in thecalculation. The large oscillations observed in the recovered groupdelay spectrum, especially towards the edges of the spectral window ofFIG. 22B, are due to the significant drop of the intensity of thereflection coefficient amplitude at those wavelengths, which makes therecovery of spectral phase more difficult. In the limiting case wherethe intensity goes to zero, the definition of phase has less meaning.However, this behavior is inconsequential, since the group delay in themore important range of wavelengths (e.g., between approximately 1548nanometers and approximately 1552 nanometers, for the target FBG isrecovered quite well. The results of FIGS. 22A and 22B show that certainembodiments described herein work well even with fairly noisymeasurements of the power spectrum.

The noise sensitivity of certain embodiments described herein is alsoaffected by the ratio of the dummy input laser pulse to reflectionimpulse response amplitudes. In certain embodiments in which the mainsource of noise in the OSA measurement system is proportional to theinput intensity, a larger dummy pulse results in a larger noiseintensity in the measured spectrum, and thus a larger error in therecovered FBG spectra. To maximize the accuracy of the recovery withnoisy measurements, certain embodiments advantageously select anamplitude ratio close to the critical ratio. For example, for the powerspectrum shown in FIG. 21, an amplitude ratio of 27 was used, which isclose to the critical ratio of 26 for the longer laser pulse, as shownin FIG. 20. In certain embodiments, this choice of the amplitude ratioensures both accurate convergence of the iterative error-reductionmethod and reduced sensitivity to measurement noise. In certainembodiments, the critical ratio that ensures convergence is traced bychoosing two different ratio values and comparing the difference betweenthe recovery results. If the difference is small, then convergence isachieved and the chosen ratio values are somewhere on the lower baselineof the curve in FIG. 20.

Certain embodiments described herein are conveniently used tocharacterize any FBG spectra uniquely. Certain embodiments also haveadvantages with respect to currently existing techniques, e.g.,low-coherence spectral interferometry. Certain embodiments describedherein provide better resolution (e.g., by a factor of two) than doeslow-coherence spectral interferometry using the same measurementconfiguration. In low-coherence spectral interferometry, by filtering inthe inverse FT domain, the convolution of the impulse response with theautocorrelation function of the input laser pulse is recovered. However,in certain embodiments described herein, the convolution of the sameimpulse response with the input laser pulse itself is recovered, whichconstitutes an improved resolution by approximately a factor of 2.

In certain embodiments described herein, the time delay between theinput laser pulse and the reflection impulse response of the FBG is madeas small as possible, as long as there is no temporal overlap betweenthe two pulses. However, low-coherence spectral interferometry requiresa certain minimum delay between these two pulses to ensure individualfiltering of the above-mentioned convolution term in the IFT domain. Incertain embodiments in which the OSA has low resolution, the large timedelay needed by low-coherence spectral interferometry can result inrapid fringes in the power spectrum that the OSA cannot resolve, whichcan potentially cause severe recovery errors.

Certain embodiments described herein are used to characterize multipleFBGs concurrently using a single OSA measurement. In certain suchembodiments, the measurement configurations of either FIGS. 7A and 7Bare used, with the additional FBGs to be characterized added in aparallel fashion next to the first FBG. FIG. 7C schematicallyillustrates an exemplary embodiment in which multiple FBGs 540 a, 540 b,540 c, . . . are coupled to a 1×N coupler 590 where N is two or more.Each of the FBGs 540 a, 540 b, 540 c, . . . are coupled to the couplerby a corresponding optical path 592 a, 592 b, 592 c, . . . each having adifferent optical pathlength. In certain embodiments, the opticalpathlengths of the optical paths 592 a, 592 b, 592 c, . . . are selectedto avoid having the reflected pulses from the FBGs 540 a, 540 b, 540 c,. . . overlapping temporally when they arrive at the OSA 570.

As described above for the measurement configuration for a single FBG540, the reflection impulse responses of all the FBGs 540 a, 540 b, 540c, . . . are time-delayed with respect to the stronger dummy laserpulse. The pulse sequence of such embodiments consists of more than 2pulses, and the pulse sequence is sent to the OSA for the power spectrummeasurement. The processing of the measured power spectrum in certainsuch embodiments is done in the same manner as described above (e.g.,using the iterative error-reduction method 700 of FIG. 9). Theconfiguration of multiple FBGs schematically illustrated in FIG. 7C isalso used in certain embodiments with the measurement configuration ofFIG. 8 to characterize multiple FBGs concurrently.

FIG. 23 is a plot of the normalized input laser pulse spectrum and thereflection spectra of two different Gaussian apodized FBGs (FBG#1 andFBG#2) which are characterized together in an exemplary embodiment. Theamplitude and phase of the temporal electric field profile of the inputlaser pulse are shown in FIGS. 24A and 24B, respectively. FIG. 25 showsthe amplitude spectrum and the phase spectrum of the pulse sequenceformed by time-delaying the two reflected pulses from the FBG#1 andFBG#2, with respect to the leading dummy input laser pulse with asplitting ratio of 40.

FIG. 26 is a plot of the power spectrum for this pulse sequence of threepulses, as would be measured by an OSA. In this exemplary embodiment,the resolution of the OSA was assumed to a modest number, e.g.,approximately 10 picometers. Using only the power spectrum measurementshown in FIG. 26, the simultaneous recovery of the reflectioncoefficient amplitude and group delay spectra for these two FBGs isachieved. FIG. 27A is a plot of the original (solid line) and recovered(dashed line) amplitude of the reflection coefficient of FBG#1. FIG. 27Bis a plot of the original (solid line) and recovered (dashed line) groupdelay spectra of the reflection coefficient of FBG#1. FIG. 28A is a plotof the original (solid line) and recovered (dashed line) amplitude ofthe reflection coefficient of FBG#2. FIG. 28B is a plot of the original(solid line) and recovered (dashed line) group delay spectra of thereflection coefficient of FBG#2.

As in the exemplary embodiments previously described, the recovery forthis exemplary multiple-FBG embodiment is very good, showing thatcertain embodiments can quite conveniently characterize more than oneFBG all at the same time using a single power spectrum measurement. Thelarge-scale oscillations observed in the recovered group delay spectrum,especially towards the edges of the spectral window shown in FIG. 27Bare due to the significant drop of the intensity of the reflectioncoefficient amplitude at those wavelength, which makes the recovery ofspectral phase more difficult. However, this is inconsequential, asdiscussed above in relation to FIG. 22B.

Certain embodiments described herein work well if any of the electricfield envelopes in the pulse sequence are time-reversed. Stateddifferently, certain embodiments described herein can advantageouslydifferentiate between a pulse and its time-reversed replica.

Certain embodiments which utilize the measurement configuration 600schematically illustrated in FIG. 8 share all the attributes describedabove in relation to the measurement configurations 500, 580 of FIGS. 7Aand 7B. Certain embodiments utilizing the measurement configuration 600advantageously provide additional desirable features over existingtechniques as well. In certain such embodiments, the dummy input laserpulse does not need to be temporally much narrower than the reflectionimpulse response of the target FBG. As described above, the measurementconfigurations 500, 580 shown in FIGS. 7A and 7B advantageously providebetter resolution (e.g., by a factor of two) with respect tolow-coherence spectral interferometry, which uses an input dummy laserpulse that is narrower than the reflection impulse response of thetarget FBG by approximately 50 times or more. However, in certainembodiments utilizing the measurement configuration 600 of FIG. 8, thesame FBG can be characterized with an input laser pulse that is onlyapproximately 2 to 5 times narrower than the reflection impulse responseof the FBG. Certain such embodiments advantageously eliminate therequirement for an ultra-short input laser pulse, which are generallymore costly and more difficult to generate than longer laser pulses.

FIG. 29 is a plot of an FBG reflection spectrum (solid line) and thenormalized input laser pulse amplitude (dotted line) in an exemplaryembodiment. The amplitude and phase of the temporal profile of the inputlaser pulse shown in FIGS. 30A and 30B, respectively, shows that thewidth of the laser pulse is approximately 18 picoseconds, where thetemporal width is defined as the full width at which the field reducesto 10% of its maximum value. The temporal width of the reflectionimpulse response of the same FBG shown in FIG. 29 is wider than theinput laser pulse by only a factor of approximately 2. As shown by FIG.31, the temporal width of the reflection impulse response isapproximately 37 picoseconds, while the reflected pulse width isapproximately 56 picoseconds.

Referring to the measurement configuration 600 of FIG. 8, the18-picosecond-wide input dummy laser pulse 620 is first split into twopulses. The first pulse 622 is sent to a delay line 630. The secondpulse is sent to a pulse stretcher 640, which broadens the temporalwidth of the laser pulse to produce a time-stretched laser pulse 624which is broadened by a factor of at least approximately 2 to 5 ascompared to the second pulse before being stretched. The pulse stretcher640 of certain embodiments is a loop of single-mode fiber optic cablethat broadens the width of a laser pulse through dispersion. In thisexemplary embodiment, the input laser pulse 620 has an initial width ofapproximately 18 picoseconds, and the time-stretched laser pulse 624 isstretched in time by a factor of approximately 4, such that the width ofthe time-stretched pulse 624 is approximately 76 picoseconds, as shownin the plot of FIG. 32. The temporal electric field profile of thetime-stretched pulse 624, i.e., E_(s)(t), is chosen in certainembodiments to be wider than the dummy input laser pulse profile by afactor of at least approximately 2 to 5.

The time-stretched pulse 624 is sent through an attenuator 650, and isthen split into two weak pulses, as shown in FIG. 8. In certainembodiments, the attenuator 650 is approximately 32 dB (in power), whilein other embodiments, the attenuator 650 has other values (e.g., greaterthan approximately 20 dB). The reflected weak pulse 672 from the targetFBG 540 is no longer the reflection impulse response of the FBG 540,since the incident pulse is much broader than the impulse response ofthe FBG 540, as shown in FIG. 31. The reflected pulse 672 is theconvolution of the impulse response of the FBG 540 with the temporalprofile of the time-stretched pulse 624, i.e., h_(R)(t)*E_(s)(t). Theback reflections from the FBG 540 and from the mirror 660 (e.g., a barefiber end) are then temporally combined with the time-delayed version ofthe unattenuated initial dummy laser pulse 622, as shown in FIG. 8. Thispulse sequence 680 has a dominant peak pulse at the leading edgefollowed by two weaker reflected pulses, as shown in FIG. 33. The pulsesequence 680 is then sent to the OSA 570, which measures its powerspectrum. Assuming a typical OSA 570, with a resolution of approximately10 picometers, the theoretical power spectrum of the pulse sequence ofFIG. 33 is shown in FIG. 34.

As discussed above, the error-reduction method 600 of FIG. 9 is used incertain embodiments to process the measured output of the OSA. Becausethe input pulse sequence is close to an MPF, both E_(s)(t) andh_(R)(t)*E_(s)(t) can simultaneously be recovered using only the OSAoutput. The reflection spectrum of the target FBG is computed in certainembodiments by taking the FTs of both E_(s)(t) and h_(R)(t)*E_(s)(t),i.e.,${r(\omega)} = {\frac{{FT}\left\{ {{h_{R}(t)}*{E_{s}(t)}} \right\}}{{FT}\left\{ {E_{s}(t)} \right\}}.}$In certain embodiments, the power spectrum of the input laser pulsecovers the frequency band of the target FBG, as shown in FIG. 29. Theresult of this exemplary embodiment is shown in FIGS. 35A and 35B. Onceagain the recovery is very good. In this exemplary embodiment, aGaussian apodized FBG with an impulse response temporal width ofapproximately 37 picoseconds has been fully characterized using a dummylaser pulse that has a temporal width of approximately 18 picoseconds byusing only a single OSA measurement. The whole computation involved inthe exemplary embodiment took only a few seconds to run using MATLAB 5on a 500-MHz computer.

In certain embodiments, various ultrashort pulse-shaping techniques(see, e.g., M. M. Wefers and K. A. Nelson, “Analysis of ProgrammableUltrashort Waveform Generation Using Liquid-Crystal Spatial LightModulators,” J. Opt. Soc. Am. B, 1995, Vol. 12, pp. 1343-1362; A.Rundquist et al., “Pulse Shaping with the Gerchberg-Saxton Algorithm,”J. Opt. Soc. Am. B, 2002, Vol. 19, pp. 2468-2478) are used to modify thetemporal profile of the dummy pulse in order to achieve a true MPF forthe electric field of the pulse sequence, which can potentially improvethe recovery speed of certain such embodiments dramatically. By using atrue MPF, certain embodiments can converge in less than 5 iterations,thus cutting down the computation time to a fraction of a second, evenwhen using a relatively slow programming environment such as MATLAB 5.1.

Various embodiments of the present invention have been described above.Although this invention has been described with reference to thesespecific embodiments, the descriptions are intended to be illustrativeof the invention and are not intended to be limiting. Variousmodifications and applications may occur to those skilled in the artwithout departing from the true spirit and scope of the invention asdefined in the appended claims.

1. A method of determining a complex reflection impulse response of afiber Bragg grating, the method comprising: (a) providing a measuredamplitude of a complex reflection spectrum of the fiber Bragg grating;(b) providing an estimated phase term of the complex reflectionspectrum; (c) multiplying the measured amplitude and the estimated phaseterm to generate an estimated complex reflection spectrum; (d)calculating an inverse Fourier transform of the estimated complexreflection spectrum, wherein the inverse Fourier transform is a functionof time; and (e) calculating an estimated complex reflection impulseresponse by applying at least one constraint to the inverse Fouriertransform of the estimated complex reflection spectrum.
 2. The method ofclaim 1, further comprising: (f) calculating a Fourier transform of theestimated complex reflection impulse response; and (g) calculating acalculated phase term of the Fourier transform of the estimated complexreflection impulse response.
 3. The method of claim 2, whereincalculating the calculated phase term of the Fourier transform comprisesusing a Hilbert transformation of the Fourier transform of the estimatedcomplex reflection impulse response.
 4. The method of claim 2, furthercomprising: (h) using the calculated phase term of (g) as the estimatedphase term of (c) and repeating (c)-(e).
 5. The method of claim 4,wherein (c)-(h) are iteratively repeated until the estimated complexreflection impulse response reaches convergence.
 6. The method of claim5, wherein convergence is reached when an average difference betweenestimated complex reflection impulse responses obtained after twoconsecutive iterations is less than a predetermined value.
 7. The methodof claim 6, wherein the predetermined value is 0.1% of the estimatedcomplex reflection impulse response of an iteration.
 8. The method ofclaim 4, wherein (c)-(h) are iteratively repeated a predetermined numberof times.
 9. The method of claim 1, wherein providing the measuredreflection spectrum amplitude comprises measuring a reflection powerspectrum and taking the square root of the reflection power spectrum.10. The method of claim 1, wherein providing the measured reflectionspectrum amplitude comprises providing a previously-measured reflectionspectrum amplitude.
 11. The method of claim 1, wherein providing theestimated phase term of the complex reflection spectrum comprisesproviding an initial estimated phase term equal to a real or complexconstant.
 12. The method of claim 1, wherein providing the estimatedphase term of the complex reflection spectrum comprises providing aninitial estimated phase term equal to a measured phase term of thecomplex reflection spectrum of a second fiber Bragg gratingsubstantially similar to the fiber Bragg grating from which the measuredamplitude of the complex reflection spectrum is measured.
 13. The methodof claim 1, wherein applying the at least one constraint to the inverseFourier transform of the estimated complex reflection spectrum comprisessetting the inverse Fourier transform to zero for times less than zero.14. The method of claim 1, wherein the complex reflection impulseresponse has a known temporal duration and applying the at least oneconstraint to the inverse Fourier transform of the estimated complexreflection spectrum comprises setting the inverse Fourier transform tozero for times greater than the known temporal duration of the complexreflection impulse response.
 15. The method of claim 1, wherein thefiber Bragg grating has a known bandwidth and applying the at least oneconstraint to the inverse Fourier transform of the estimated complexreflection spectrum comprises adjusting the inverse Fourier transform toprovide the known bandwidth.
 16. The method of claim 1, wherein thereflection impulse response is a real function and applying the at leastone constraint to the inverse Fourier transform of the estimated complexreflection spectrum comprises using only the real portion of the inverseFourier transform.
 17. A computer-readable medium having instructionsstored thereon which cause a general-purpose computer to perform themethod of claim
 1. 18. A computer system comprising: means forestimating an estimated phase term of a complex reflection spectrum of afiber Bragg grating; means for multiplying a measured amplitude of thecomplex reflection spectrum of the fiber Bragg grating and the estimatedphase term to generate an estimated complex reflection spectrum; meansfor calculating an inverse Fourier transform of the estimated complexreflection spectrum, wherein the inverse Fourier transform is a functionof time; and means for calculating an estimated complex reflectionimpulse response by applying at least one constraint to the inverseFourier transform of the estimated complex reflection spectrum.
 19. Amethod of determining a complex transmission impulse response of a fiberBragg grating, the method comprising: (a) providing a measured amplitudeof a complex transmission spectrum of the fiber Bragg grating; (b)providing an estimated phase term of the complex transmission spectrum;(c) multiplying the measured amplitude and the estimated phase term togenerate an estimated complex transmission spectrum; (d) calculating aninverse Fourier transform of the estimated complex transmissionspectrum, wherein the inverse Fourier transform is a function of time;and (e) calculating an estimated complex transmission impulse responseby applying at least one constraint to the inverse Fourier transform ofthe estimated complex transmission spectrum. 20.-32. (canceled)
 33. Amethod of characterizing a fiber Bragg grating, the method comprising:(a) providing a measured amplitude of a Fourier transform of a complexelectric field envelope of an impulse response of the fiber Bragggrating; (b) providing an estimated phase term of the Fourier transformof the complex electric field envelope; (c) multiplying the measuredamplitude and the estimated phase term to generate an estimated Fouriertransform of the complex electric field envelope; (d) calculating aninverse Fourier transform of the estimated Fourier transform of thecomplex electric field envelope, wherein the inverse Fourier transformis a function of time; and (e) calculating an estimated electric fieldenvelope of the impulse response by applying at least one constraint tothe inverse Fourier transform of the estimated Fourier transform of thecomplex electric field envelope. 34.-45. (canceled)
 46. An apparatus forcharacterizing at least one fiber Bragg grating, the apparatuscomprising: a laser pulse source, the laser pulse source generating atleast one input laser pulse; an optical spectrum analyzer; a firstoptical path optically coupled to the laser pulse source, the firstoptical path comprising a pulse stretcher and an attenuator, wherein afirst portion of the input laser pulse propagates from the laser pulsesource and is stretched by the pulse stretcher and is attenuated by theattenuator; a second optical path optically coupled to the first opticalpath and comprising a mirror, wherein a first portion of the stretchedand attenuated laser pulse from the first optical path is reflected fromthe mirror; a third optical path optically coupled to the first opticalpath and comprising a first fiber Bragg grating, wherein a secondportion of the stretched and attenuated laser pulse from the firstoptical path is reflected from the first fiber Bragg grating; a fourthoptical path optically coupled to the second optical path, the thirdoptical path, and the optical spectrum analyzer, wherein the reflectedpulse from the mirror and the reflected pulse from the first fiber Bragggrating propagate to the optical spectrum analyzer; and a fifth opticalpath optically coupled to the laser pulse source and the opticalspectrum analyzer, the fifth optical path comprising a delay line,wherein a second portion of the input laser pulse propagates from thelaser pulse source along the fifth optical path to the optical spectrumanalyzer.
 47. The apparatus of claim 46, further comprising a sixthoptical path optically coupled to the first optical path and the fourthoptical path, the sixth optical path comprising a second fiber Bragggrating, wherein a third portion of the stretched and attenuated laserpulse from the first optical path is reflected from the second fiberBragg grating, the reflected pulse from the second fiber Bragg gratingpropagating to the optical spectrum analyzer.
 48. The apparatus of claim46, wherein the laser pulse source comprises a mode-locked laser with atemporal width between approximately 2 picoseconds and approximately 4picoseconds.
 49. The apparatus of claim 46, wherein the optical spectrumanalyzer has a resolution less than or equal to 10 picometers.
 50. Theapparatus of claim 46, wherein the pulse stretcher comprises asingle-mode fiber optic cable which broadens the temporal width of apulse propagating therethrough by a factor of approximately 2 to
 5. 51.The apparatus of claim 46, wherein the attenuator is greater thanapproximately 20 dB.
 52. The apparatus of claim 46, wherein the mirrorcomprises a bare fiber end.
 53. The apparatus of claim 46, wherein themirror comprises a mirrored fiber end.